Help with orbital mechanics for an unusual alien planet

I am embarking on creating a new gaming campaign world, something I've wanted to do for years but never quite manged: an entirely alien world (i.e. not populated by humans or the like), yet with the same sort of technology as you'd get in "typical" fantasy (with maybe a few sci-fi elements thrown in.) My basic idea, though, is sufficiently ambitious as to require so considerable thought into the orbital mechanics of system - and I am the sort of pedant who can't just hand-wave away stuff because it's cool. So I am trying to work out the basic orbital mechanics to at least a reasonable level of plausibility. (So that it at least may be said "he at least tried to do the research!") My intention is to try and start out with as reasonable an approximation as I can manage before having to put in too much "a wizard/exotic material (etc) did it" handwaving. I have broached the topic in my usual forum haunts (which for a kick-off gave me some ideas of the sort of things I needed to read up on, but wikipedia only takes one so far), but I thought it might be worth finding some additional knowledgeable minds to talk to. (I am not an astrophysist, though I do have some engineering training, so I'm not completely clueless mathmatically, just so we know where I stand!)

The high-end concept for the world was "the evenstar" a world of perpetual evening (or at least lit like it it was perpetual twilight) that every so often would be plunged into true night.
There would thus be a (great?) number of fallen civilisations that went extinct during previous
incidents, but that the natural ecology of the planet would be "used" to it. (I'm drawing partial
inspiration from what I think was a Doctor Who episode (Pertwee or Tom Baker era, I hazard) where basically the planet's life - including the people - mutated into new forms in some period because of the planet's rotation or something.) Anyway, the idea was that there would be this perodic disaster that knobbled the people without setting the planet's ecology back like a true mass extinction would.

While one way to do this would have been just to do a very long day/night cycle, I decided that was too "obvious" from the perspective of the natives on the planet (and also, you'd have serious climate issues during the "nigh" periods.) I wanted the event to be a bit more mysterious than that. I thus settled on the idea of using a tide-locked planet (or the terminous region anyway) that would give you the "evening"-ish bit, but that then required finding some way of making the it "night." After bouncing ideas around, someone finally suggested used a modified RCB type- star (e.g. R Coronae Borealis) where the luminousity dims every so often.

Currently, my working premise is thus:

A planet tide-locked to a red dwarf star (and moderately distant from it), which orbits an RCB
variable star. Most of the light (and presumably, heat) comes from the RCB star, except during
periods when the luminosity dims. The red dwarf keeps the planet's heat ticking over, as if were,
during the occlusion periods (like putting the oven on to keep something warm), provides some dim light. (My assupmtion is it's going to be a bit like Gliese 667C and 667Cc in the there's less light, but equal or more infrared.) The RCB star is slightly unusual by even RCB standards, having a long burn, with extended periods between its dark periods that are about twos orders of magnitude longer than "usual". I will assume, fo the sake or argument, that the RCB results from, as theoried as possible, the collision of two white dwarf stars, previously a stellar binary. I nominally postulate that perhaps a greater distance from the red dwarf (by habitability standards, which is really, really close!) might reduce some of the problems caused by early flaring, though by this point in the system's life-cycle, you might expect it to have past that stage anyway. (And also perhaps reduce some of the visible light for aethetic effect.)

Drawing from Aurelia, the planet speculated upon in the TV series Alien Planet, I'm postulating the
sunward side is covered by a continunous rainstorm, and the dark side is kept from freezing as
suggested in my reading by the air currents.

Purely because during my wiki reading around the subjects I happened across he artist's impression of Gliese 667 Cc on wikipedia, I am going to assume the RCB star/ companion star are also part of a multistar system that contains two other binary stars in the distance. (Essentially, at one point it would have been a system with two pairs of binary stars, one pair of which had a companion red dwarf star.)

A kind gentleman on my regular rolplaying forum was good enough to do some scratch calculations for me as a starter for ten, which I shall include as posted.

Anyway, here's an attempt at some physical properties for this system. Mass, luminosity, and radius are in multiples of the values for the Sun; distances are in AU. I've heavily based the

Planet
Orbits Secondary at ~0.0345 AU (period ~178 hours)
Ice caps from ~90 to 140 degrees west of substellar point, at +-45 degrees of latitude.
Water flows form the ice caps, evaporates as it travels to the day side.

Secondary delivers about 1000 W/m2 to Planet, while Primary delivers about 370 W/m2 (for comparison, the Sun delivers ~ 1366 W/m2, so the combined total is roughly the same).

Primary appears to rise and set every 7 1/2 days due to the rotation of Planet.

Primary has an angular diameter of about 1/4 of a degree as seen from Planet (about half the angular diameter of the Sun or Moon as seen from Earth), while Secondary has an angular diameter of about 4 degrees.

Realistically, Secondary's orbit would be eccentric, so there would also be some predictable
variability in the amount of light it delivers, with a period of around 5000 years.

Flares can make the light side of Planet uninhabitable, although that can probably be achieve by
temperature alone (and it might be worthwhile to have the light side be marginally habitable).

If you want high vulcanism, add a gas giant or two in resonance with Planet to drive its orbital
eccentricity up.

First thing that occurs in that is that if the dwarf star is not itself tide-locked to the RCB star, there will presumably be a period when the primary is set (when the planet is between the two stars) and you might hard-pressed to tell the difference between that "night" and the one caused by the dimming of the primary. (Though that effect might be reduced in the orbital plane of the dwarf/planet system is "below" the level of the orbital plane of the dwarf/RCB star, at least on part of the planet...?) Yes? No?

Ideally, I would like to add some volcanisity, which as the gentleman suggested would require a gas giant in the "vicinity?" If is was a reflective gas giant, might this also go some way to meaning that the orbital period of the secondary around the RCB star is not as dark as the periods when the primary's light dims?

I would also like to include at least one moon. I am thinking of a small, irregular-shaped moon (presumably much closer than Luna and dramatically smaller, more sort of Phobos or even smaller). I am uncertain as to whether this would have to be tide-locked, or whether it could be in orbit. (I do like the image of a rough blocky moon seen even in the "day." Blame Stewart Cowley's TTA books for burning those sort of images into my mind at an impressionable age...!)

I am also toying with the idea that the planet should not be quite tide-locked, but has a very slow day/night cycle (i.e. thousand plus years), which presumably would also have some effect on the volcanicity (as well as providing for come ancient civilisations to have moved into the dark side.)

My next problem is going to be time measurement. Obviously, with no day/night cycle, there's not
going to be a "day" but you sort of need a unit of time of about that long. There's not a "year" either, strictly speaking, so determining where you'd start working time from is an interesting question. I also am fairly sure there would be no seasons, either, at least save very long-term.

I am considering "waning:" the period of time in which the primary RCB appears to rise and set. So, while not exactly a "night" because of the secondary, there would at least be (In the figures above, that would be 178 hours. If I cheat a bit and call it 180, you could have a
"waning" being a week of six thirty-hour segments). (I am sort of assuming that given the inevitable slight eccentricitys of orbital systems, that the secondary may past close to, but not necessarily eclipse the primary during it's rotation, at least not on a regular basis.) Depending on whether or not you could have an orbiting moon (or whether it'd have to be tide-locked), that period might be able to be worked in as maybe something to do with "day" or "year" or something. (I am guessing that the apparent orbital period of the distant binaries would be very long, thousands of years possibly.)

Suggestions and observations would be greatly welcomed on any and all of these points - and also any other suggestions or ideas these environmental structures might have on the life that would emerge (I've not quite got that far yet, since I wanted to start at the very basics first!)

Just a few comments for the time being:
1.Your "working premise" as you describe it is at odds with the setup the "gentleman" provided. That is, one says the bulk of solar flux comes from the primary, the other says it's from the secondary. Which one you actually want?

2.At ~270 AU tidal locking of the secondary witht the primary is implausible. The tidal torques would be tiny at that distance(tidal forces fall with the third power of distance from the source), and massive objects posses a lot of angular momentum that needs to be altered. I'd rather ditch that idea.

3.Since the planet has got slow rotational speed of ~7 1/2 days, any satellites need to have at least the same period(case of a tidally locked satellite), or longer. Otherwise the tidal interactions would act to pull them towards the planet eventually ending up in a collision.
7 1/2 days period corresponds to no less than about half the Earth-Moon distance(from Kepler's 3rd).

4.The 7 1/2 day(or any other) orbit around the secondary would mean the primary would rise and set with that period. You can't really have it perpetually shine(until it dims) over any given spot on the planet. Apart from the poles, that is, but then again it's very low on the horizon there.
So, you do have normal-ish days(i.e., evenings) and nights, only sometimes, when it dims, the primary appears not to rise.

5.There's an interesting paper about tidally-locked exoplanet climates:Stabilizing Cloud Feedback Dramatically Expands the Habitable Zone of Tidally Locked Planets
which suggests that such planets can have relatively low temperatures despite as high as 2200W/m^2 incident solar flux. This means you could conceivably put the planet closer to the secondary, increasing the revolution period, and giving shorter 'days' as given by the visiblity of the primary.
It's an interesting read, that might give you some ideas regarding the expected cloud cover etc. There's a lot more to be found on arxiv.org if you type "tidally locked habitability" etc.

6. I remember reading a paper analysing the impact of red dwarf UV flaring on habitability. IIRC, it concluded that it's not a problem, as the total UV output during flares for such cold stars merely reaches quiescent levels we get from the Sun. I could dig it up if needed be.
Anyway, I don't think you should worry about this bit too much.

Just a few comments for the time being:
1.Your "working premise" as you describe it is at odds with the setup the "gentleman" provided. That is, one says the bulk of solar flux comes from the primary, the other says it's from the secondary. Which one you actually want?

...

Good question!

From the quoted stats I think I was working on the assumption that the secondary would be delivering the most heat (i.e. infrared) but less visible light. I don't know whether that's really possible though, now you mention it.

Hmm. On the one hand, the light from the secondary is going to be constant; which is good for the "twilight lighting" effect, but bad from the "it's night" aspect. But if the bulk of the heat and light comes from the primary, we're back to point 4 - the day/night problem.

Bandersnatch said:

4.The 7 1/2 day(or any other) orbit around the secondary would mean the primary would rise and set with that period. You can't really have it perpetually shine(until it dims) over any given spot on the planet. Apart from the poles, that is, but then again it's very low on the horizon there.
So, you do have normal-ish days(i.e., evenings) and nights, only sometimes, when it dims, the primary appears not to rise.

Yeah... sort of defeating the point of having the planet tide-locked to the dwarf star in the first place to fix the light levels, doesn't it...? At that point you begin to wonder if you shouldn't have the planet just tide-locked to the primary in the first place (except that I'm guessing that being close enough to be ticde-locked to an RCB star is not going to be within the habitability zone...)

Guh. This is really hard. About the only other thing I can think of is if the secondary and planet are orbiting "below" the primary (i.e. like a pendulum spun on a string) - presumably by influence of another star - meaning you'd have the "upper" hemisphere alays in "day" from the primary and the "lower" always in "night" in addition to the day and night side of the primary. But that's a) seeming a bit implausible and b) is likely to give you the same habitability issues as tide-locking (maybe lesser if the primary is distant...?) meaning the "upper day" parts and "lower night" parts would be unhabitable, and the habitable terminator would be sort of a diagonal stripe... Which might work I suppose, but seems like it might be way complicated.

You could, I suppose, just having a really low orbital period around the secondary, giving you a really long day/night cycle again, with the dwarf star really only there to provide a source of heat to keep the planet from freezing too much during the extended night period. But that doesn't sound very satisfactory, somehow.

Bandersnatch said:

2.At ~270 AU tidal locking of the secondary witht the primary is implausible. The tidal torques would be tiny at that distance(tidal forces fall with the third power of distance from the source), and massive objects posses a lot of angular momentum that needs to be altered. I'd rather ditch that idea.

And again, I'm guessing if the dwarf star is close enough to it to be tide-locked to the primary it's going to be too close to the primary for habitation. (Is the tide-lock distance a factor of the primary or partly of the body it's acting upon? I.e., would the tide-lock distance for the dwarf star around the primary be larger than the just the planet alone around the primary?)

Bandersnatch said:

3.Since the planet has got slow rotational speed of ~7 1/2 days, any satellites need to have at least the same period(case of a tidally locked satellite), or longer. Otherwise the tidal interactions would act to pull them towards the planet eventually ending up in a collision.
7 1/2 days period corresponds to no less than about half the Earth-Moon distance(from Kepler's 3rd).

So an orbiting satelite around the planet could be slower than 7 1/2 days? Which could give you a "month" or "year" interval, maybe?

A small, close moon is plausible then.

Bandersnatch said:

5.There's an interesting paper about tidally-locked exoplanet climates:Stabilizing Cloud Feedback Dramatically Expands the Habitable Zone of Tidally Locked Planets
which suggests that such planets can have relatively low temperatures despite as high as 2200W/m^2 incident solar flux. This means you could conceivably put the planet closer to the secondary, increasing the revolution period, and giving shorter 'days' as given by the visiblity of the primary.
It's an interesting read, that might give you some ideas regarding the expected cloud cover etc. There's a lot more to be found on arxiv.org if you type "tidally locked habitability" etc.

Useful to know (though off the top of my head I can't think of a way that would help the whole day/night problem, though it could help the "day time period length" one).

I will bear the site in mind for some further reading as well, though, thanks!

Bandersnatch said:

6. I remember reading a paper analysing the impact of red dwarf UV flaring on habitability. IIRC, it concluded that it's not a problem, as the total UV output during flares for such cold stars merely reaches quiescent levels we get from the Sun. I could dig it up if needed be.
Anyway, I don't think you should worry about this bit too much.

Yeah, I had figured that was probably the leat of the problems - even the stuff I'd read on wiki suggested that the flaring would probably only last a relative short period compared to the life of dwarf star, and I was assuming this would an an old system.

Bandersnatch said:

7. I'm not sure what do gas giants have to do with volcanism.

I'm not quite sure myself, but that was what was suggested (in the last line of the quote from my associate). So, failing that, would there be much volcanicity on a tide-locked (and probably very old) planet?

If not, I think there definitely needs to be a very slow (thousand year plus) rotation of the planet relative to the primary (so it's not quite tide-locked) so as to have some of the older civilisations drift into the permenant night zone.

Okay. I've had another long think about it, and I think I may have a solution to the "day/night" cycle problem (going back to something someone suggested when I posted it on my regular forum).

If we put a reflective body (either a moon or maybe make the planet a moon of a reflective gas giant) tide-locked between the planet and the secondary, could that bounce enough (for the sake of arguement) moonlight to make the "night period" (that is, when the primary is set) bright enough to be still nearly day?

I'm sort of envisioning is as the (modal) angles being such that in the sky (for visualiation purposes) the three bodies would be slightly offset, so that, from the planet surface, at mid-primary-day, you have the moon (which would be fairly dark, but presumable not completely dark if light bounces back from the planet to the moon and back...?), then the secondary "above" it in the sky, with the primary "above" that.

The "night" would be cooler, but still bright. The moonlight would fade when then periods of occulsion of the primary show up, giving you true "night."

I would assume the reflective body would be apparently bigger than out moon, and/or more reflective. (If if was a physical moon, maybe it's made out of predominantly silver or something...! Or if a gas giant, have a white atmosphere.)

From the quoted stats I think I was working on the assumption that the secondary would be delivering the most heat (i.e. infrared) but less visible light. I don't know whether that's really possible though, now you mention it.

Sure, that sounds plausible. I was merely pointing out that there was a lack of clarity as to which body was supposed to be delivering the bulk of wattage.

At that point you begin to wonder if you shouldn't have the planet just tide-locked to the primary in the first place (except that I'm guessing that being close enough to be ticde-locked to an RCB star is not going to be within the habitability zone...)

Well, I'm thinking it might be not a completely bad idea.
The thing is, the amount of angular momentum a planetary body ends up with after most of the protoplanetary debris had been cleared during the formation of the system can be almost anything, especially for smaller bodies(i.e., not gas giants). So you could conceivably have a planet that never really had much of angular momentum to shed, so it eventually did get tidally locked to the distant star, especially if given a lot of time.
In this scenario, there is no secondary, the primary is at about half the distance specified earlier(~135AU should net a healthy ~1400 watts/sq.metre of insolation)
You can't have a large moon here, though. It would act to un-lock the planet from the star, and at such a low rotation(1 day=1 year=~1600 Earth years) any moon that lies within the planet's hill sphere(http://en.wikipedia.org/wiki/Hill_sphere) would end up spiralling downwards to a crash.
You could conceivably imagine one or a few tiny rocks as recent captures. They'd be too small to induce much torque on the planet. Maybe some libration(http://en.wikipedia.org/wiki/Libration) at worst, leading to the star to dance in the sky around a point in the sky. They would eventually crash, though, but it could be in such a timescale so as to be irrelevant to the peoples living there.

The problem is, I know almost nothing about the evolution of RCB stars. A quick search suggests that nobody really knows much about them(I might be wrong).
If you assume they're the result of white dwarfs merger, then it was most likely a rather catastrophic event, not boding well for any life in the system. Not to mention the deaths of whatever stars produced the white dwarfs. And if it was catastrophic, was there enough time for new life to develop while the RCB star is in its stable phase?(it's a supergiant, so I reckon it's not going to live long).

Then again, since the type of stars is so poorly understood(I think), you could just use your licentia artistica to fill the blank spaces in knowledge without damaging your SF-hardness score.

And again, I'm guessing if the dwarf star is close enough to it to be tide-locked to the primary it's going to be too close to the primary for habitation. (Is the tide-lock distance a factor of the primary or partly of the body it's acting upon? I.e., would the tide-lock distance for the dwarf star around the primary be larger than the just the planet alone around the primary?)

As mentioned earlier, it's not that implausible.
The tidal locking is the result of tidal forces producing torque on the orbiting body. The forces scale linearly with mass of the orbited body and the satellite's size, but are very strongly inverselly dependent on the distance between the two(##\propto{\frac{1}{R^3}}##). The amount of torque needed is dependent on the moment of inertia of the satellite - less massive ones are easier to lock - and the initial angular momentum.
If the initial ang.momentum is small, even low torques on large bodies can do the trick if given enough time and no other nearby bodies to mess up the process.

Going back to the binary system scenario:

So an orbiting satelite around the planet could be slower than 7 1/2 days? Which could give you a "month" or "year" interval, maybe?

A small, close moon is plausible then.

Yeah. Just remember that the farther the moon you place, the slower its orbital period.
You said you have an engineering background, so you should be able to work the details out for a given period or distance(the force of gravity needs to equal the centripetal force, or just use Kepler's 3rd law).

Yeah, I had figured that was probably the leat of the problems - even the stuff I'd read on wiki suggested that the flaring would probably only last a relative short period compared to the life of dwarf star, and I was assuming this would an an old system.

Actually I think I had misunderstood you. I thought you were talking about UV flares on the red dwarf secondary(which will happen a lot). Now I see you were thinking of the white dwarf progenitors merging. As mentioned earlier, it sounds like it could be a problem for the life on the planet. But. I don't think the WD merger is the definite hypothesis as to the origins of RCB stars, so you could always handwave the origins and assume it wasn't anything harmful.

So, failing that, would there be much volcanicity on a tide-locked (and probably very old) planet?

I think it would be pretty damn quiet. But, you could have another planet orbiting the secondary nearby, in some orbital resonance, leading to tidal forces varying radially over time and inducing volcanism(something like with Jupiter's Io).
Perhaps that's what the gentleman meant by introducing a gas giant.
Get's a bit harder to justify with just the RCB star scenario.

If not, I think there definitely needs to be a very slow (thousand year plus) rotation of the planet relative to the primary (so it's not quite tide-locked) so as to have some of the older civilisations drift into the permenant night zone.

It would fit well with the "just RCB star"-scenario. The planet would need to orbit the star with about 1600 year period, and if it wasn't fully locked, it could have almost any period of "day/night" cycle(speaking of the RCB star raising and setting, not the dimming).

If we put a reflective body (either a moon or maybe make the planet a moon of a reflective gas giant) tide-locked between the planet and the secondary, could that bounce enough (for the sake of arguement) moonlight to make the "night period" (that is, when the primary is set) bright enough to be still nearly day?

I'm not going to tackle this one right now, as my head starts spinning. It looks like a mess of tidal interactions that I'm not entirely sure would produce anything remotedly stable. I might try to work it out later on.

Sure, that sounds plausible. I was merely pointing out that there was a lack of clarity as to which body was supposed to be delivering the bulk of wattage.

Well, I'm thinking it might be not a completely bad idea.
The thing is, the amount of angular momentum a planetary body ends up with after most of the protoplanetary debris had been cleared during the formation of the system can be almost anything, especially for smaller bodies(i.e., not gas giants). So you could conceivably have a planet that never really had much of angular momentum to shed, so it eventually did get tidally locked to the distant star, especially if given a lot of time.
In this scenario, there is no secondary, the primary is at about half the distance specified earlier(~135AU should net a healthy ~1400 watts/sq.metre of insolation)

Okay.

Bandersnatch said:

The problem is, I know almost nothing about the evolution of RCB stars. A quick search suggests that nobody really knows much about them(I might be wrong).
If you assume they're the result of white dwarfs merger, then it was most likely a rather catastrophic event, not boding well for any life in the system. Not to mention the deaths of whatever stars produced the white dwarfs. And if it was catastrophic, was there enough time for new life to develop while the RCB star is in its stable phase?(it's a supergiant, so I reckon it's not going to live long).

Then again, since the type of stars is so poorly understood(I think), you could just use your licentia artistica to fill the blank spaces in knowledge without damaging your SF-hardness score.

Yeah, I hadn't heard of them myself until someone mentioned it to me; there just doesn't appear to be much information on them (they've only found about a hundred, so wiki tells me). So the grey area gives me a bit of room to play with, and say this is one that has much slower "burn" than the typical ones. (Maybe because of some exotic material in the star's composition.)

The white-dwarf theory is apparently only one possibility. I thought about that since, with the extremely long-life of a red dwarf star, I was working on the basis that life on the planet may have come AFTER the cataclysmic event that lead to the RCB star... But having a closer look, that suggests than even if the star is burning "slower" than R Coronae Borealis (interval of a few months to a few centuries = 1300 times "slower"), you'd still only be in a handful of millions of years territory. (R Coronae Borealis is preportedly a yellow supergiant, which normally lasts a few thousand years (estimate 5000-ish? (On the basis that 10k plus would be reported as "tens of thousands."))

Actually, that's a bit problematic all itself, since 6.5 million years is not all that much time for even the life to adapt drastically, isn't it? (At least not with as massive a diversification as you'd expect, given that there would likely have been mass extinction events when the dimming cycle started, regardless of cause.)

Hmm. So the RCB star needs to be burning about a something like 10-20 thousand to a million times "slower" than a regular main sequence yellow supergiant (i.e. giving you 50-100 million years to a few billion for life to either adapt sufficiently widely or evolve completely, and with luminoisty intervals about a thousand times slower. Interesting...

(I think "exotic materials in structure" might have to be part of the explanation...!)

Bandersnatch said:

As mentioned earlier, it's not that implausible.
The tidal locking is the result of tidal forces producing torque on the orbiting body. The forces scale linearly with mass of the orbited body and the satellite's size, but are very strongly inverselly dependent on the distance between the two(##\propto{\frac{1}{R^3}}##). The amount of torque needed is dependent on the moment of inertia of the satellite - less massive ones are easier to lock - and the initial angular momentum.
If the initial ang.momentum is small, even low torques on large bodies can do the trick if given enough time and no other nearby bodies to mess up the process.

So you could have the secondary tide-locked to the primary, and the planet tide-locked (in the Hill radius) of the secondary?

I'm not sure what help that is to this situation, mind...! (Though you'd at that point have stable light-levels, aside from the primaries dimming period...)

Bandersnatch said:

You can't have a large moon here, though. It would act to un-lock the planet from the star, and at such a low rotation(1 day=1 year=~1600 Earth years) any moon that lies within the planet's hill sphere(http://en.wikipedia.org/wiki/Hill_sphere) would end up spiralling downwards to a crash.
You could conceivably imagine one or a few tiny rocks as recent captures. They'd be too small to induce much torque on the planet. Maybe some libration(http://en.wikipedia.org/wiki/Libration) at worst, leading to the star to dance in the sky around a point in the sky. They would eventually crash, though, but it could be in such a timescale so as to be irrelevant to the peoples living there.

Bandersnatch said:

Going back to the binary system scenario:

Yeah. Just remember that the farther the moon you place, the slower its orbital period.
You said you have an engineering background, so you should be able to work the details out for a given period or distance(the force of gravity needs to equal the centripetal force, or just use Kepler's 3rd law).

So, in order to have a proper moon (even a pretty small and close one) - i.e. one that's been there for more than a few million years, you'd need to have the secondary star system as opposed to just the RCB/planet system?

Actually, looking at, say Phobos, that's got about fifty million years left to go, which is probably "enough" time.

What I probably need to do then, is work out what apparent "size" I want the moon (given that Luna is about 30').

Now, I understand the concept of stradian measurement. I once tried my hand at writing an RPG system. The only thing of note really was I attempted to measure size and range using a stradian approximation. Basically, that everything has an apparent size to the viewer, and that if you double the viewing distance, you halve the dimensions (quarter the apparent area). (So what it meant was, instead of having a range band per se, a far target was basically just a smaller size category.) I number crunched it to within what I considered to be not far off for the Moon (i.e. what it looked like in size to me, kinda). (Bit of a crude approximation, but it was only a RPG system at the end of the day, where working out the to-hit penalty for shooting the moon was... probably never going to come up...!)

So really all I need to do is get my head around using the proper units - especially given astronomy (i.e. arcminutes) and find the right formula for calculating arcminutes from radius and distance and I can fiddle with working out how big the moon could be and how far out it is. And from that work out the orbital period.

[STRIKE]The right formula appears to be

tan (Θ (angular size)) = x (moon actual diameter)/ d (distance)

if I'm reading wiki properly. [/STRIKE]

(Or at least as close an approximation as I think I need to make - the redshift function is making my head spin rather too much! (It has been about fifteen years since I had to do complex maths when I did my engineering degree, I'm a bit rusty...!))

Edit: Or at a pinch, I could save myself some faffing about and use an online calculator...! *skullpalm* (Still, I needed to read up so as to twig tan (Θ) = degrees for plugging in the formula... And then I realised there's calculator for that too! This sort of thing just wasn't floating around the net when I first got net access when I was doing this sort of thing fifteen years ago!)

Bandersnatch said:

I think it would be pretty damn quiet. But, you could have another planet orbiting the secondary nearby, in some orbital resonance, leading to tidal forces varying radially over time and inducing volcanism(something like with Jupiter's Io).
Perhaps that's what the gentleman meant by introducing a gas giant.
Get's a bit harder to justify with just the RCB star scenario.

I suspect that's what he meant, yes.

I'm guessing that a moon big enough to cause eccentricites would also be big enough to screw up the near-tide lock?

Edit: Thought? Would the presence of the distant other pair of (otherwise purely aethetic) binary stars be enough to give some eccentricity?

Bandersnatch said:

It would fit well with the "just RCB star"-scenario. The planet would need to orbit the star with about 1600 year period, and if it wasn't fully locked, it could have almost any period of "day/night" cycle(speaking of the RCB star raising and setting, not the dimming).

Right, good!

Bandersnatch said:

I'm not going to tackle this one right now, as my head starts spinning. It looks like a mess of tidal interactions that I'm not entirely sure would produce anything remotedly stable. I might try to work it out later on.

Hah! No problem. Never let it be said I don't make things over-complicated! I can never do things the simple way...

That is, drop the tangens function and simply use ##θ=\frac{d}{r}##, where θ is the angular size in radians, d diametre and r distance. There's really no reason to bother with the precise formula, as the approximation error is going to be negligible for your needs. You'd get just 1% error for 10° angular size.

Okay, sat down today to take a proper stab at some number crunching. As being of an engineering mindset (if not one by profession, it comes from the training and family tendancies), I think best with my pencil/word document, so this is partially a stream of consciousness (largely presented hee so someone can tell me if I'm making huge and stupid mistakes in the reasoning!)

As I have my scientific calculator sat by my PC at all times, am a stickler for needless accuracy, I'm going to go ahead and use the Tan θ anyway.

I also found a website called orbitsimulator.com, which has - to my great delight and conveniance, several ready-built calculators, the first one of which I'll be using is one for the Hill Sphere.

For the sake of arguement, I looked at having a reflective gas giant, that would appear to be twice the size of the moon (working on the basis of twice apparent sixe = more reflected moonlight), just as a starter for ten.

(It took me two attemps to get to this stage, when I realised I'd somehow managed to get the moon density out by a factor of 100 when converting g/cm³ to kg/km³. Just goes to show, online calculators are useful, but you still need to be careful! I suspect given the density calculator I was using had many different conversions, I just selected the wrong line!)

Okay. I'm guessing by the time you get to that stage, the moon's gravity on the planet is going to be non-trivial.

...

There's no real getting around this, really, is there? A combination of ambitiousness and pendantry is going to require it...

How would you calculate (or approximate) tidal forces? (I mean, more specifically than "with great difficulty!")

Side query is the L1 related to tide-locking somehow?

Looking at the aformentioned orbitsimulator.com, there was a calculator for inner and outer reach (the distance inside and outside the planet's orbit at which the planet's gravity makes it unstable for other planets).

(It did require me to do a bit of reading on the paper (which is on determining exosolar planet habitability, specifically about Earth-sized planets being inside the HZ but out of destabilising gravitational influence of giant exosolar planets) to determine what value of n to use (from that work, if the orbit eccentricity is 0 or near 0, it's 3 - that's probably suitable enough for my purposes!))

Does that sound like that would be of relevance?

That is, working out the inner and outer reach of Andorlaine and putting the moon at an orbital radius of outside those values? If my limited understanding of the working is right, that would put it in a ball-park figure of being at least in a stable orbit.

Edit: Maybe: using Earth (with 0 eccentricity, granted), the interior and exterior reach came out at [STRIKE]1.45×106 and 1.54×106, which means roughly +/-44900 km from the planet's orbit (so 38500km from the planet's surface (well below the orbit of the moon). (Is the fact it's about 10% of the orbital distance of the moon just coincidental..?)So... it might be possible![/STRIKE]

Edit edit or the above thought is cobblers, because I put the Hill sphere not the orbital radius in...!

So, it's still less than Mars/ more than venus (2.39×108 km/ 1.08×108 km), which is means (given as I check with Andorlaine) the reach distances are now consistently a factor of 100 larger than the Hill sphere, so I must be doing something right...!)

Given that Andorliane's inner/outer reach is 652900000 km, to appear twice the apparent size of Luna and out outside that reach, a planet would have to be 81.4 times the size of Jupiter. Erm. Yeah. I... don't think I'm on the right track here, somehow, have for calculating how far away other orbiting bodies are.

Hehe, I know what you mean about having a pencil and paper on hand! :)

I'm not sure you quite got what the Hill sphere is all about. Or maybe I'm just missing some part of your reasoning.
Why did you calculate all those Hill spheres for the various aesthetic satellites? It's of no consequence. All you need, is the Hill sphere of the planet with relation to the star - it's the region around the planet where it is sensible to place satellites, as they can be expected to orbit the planet rather than just the star. Farther than that and the stellar gravity plucks the satellite from the planet's grasp.
Doing the same for those satellites, you find the radii at which an object would orbit those satellites. You don't need that.

So, you need just the first of the numbers you calculated. It's meaning is that you can't have a satellite farther than that.

How close it can be, is directly connected with the fact that the planet is tidally locked(or nearly so), and as such(at such great distance) will have very slow rotation.
How slow the rotation is, in turn, determines the evolution of the orbits of the planet's satellites. If the period of the (sat's)orbit is faster than the SLOW rotation of the planet, it will tidally interact with the planet in such a way so as to spiral down and eventually crash onto the surface*.

So, you need to calculate the geostationary orbit distance(satellite's orbital period=planet's rotation). This is the lower limit. The lowest orbit that won't see the satellite crash onto the planet some day.
But. It could take a LONG time, so take comfort in that. Think our Moon in reverse - it was once very much closer and it took it billions of years to move to where it is now, so you could just disregard it, unless you want to incorporate a doom prophecy into your writing or something.

(You should be able to do the calculations if you really need to. It's a secondary school-level exercise of comparing the gravitational force with the centripetal force; but do give us a shout if you end up stuck)

What I meant to say when I first mentioned the Hill sphere, was that the geostationary orbit for a planet with a rotational period in the vicinity of a thousand+ years may well lie beyond the planet's gravitational influence(the Hill sphere).

*If you care to visualise it, it works like this: the two orbiting bodies create tidal bulges on each other. Let's disregard the bulge on the satellite(m<<M). The bulge on the planet may get displaced so that it does not lie exactly on the line connecting the two bodies - displacement ahead of rotation if the rotation of the planet is faster than than of the satellite; displacement lagging behind if the rotation is slower.
In the first case, the displaced bulge exerts tangential pull on the satellite in the prograde(direction of motion) direction, adding energy to its orbit and causing it to spiral away(like our Moon does).
In the second case the lagging bulge pulls retrograde, sapping satellite's orbital energy and making it spiral downwards.

For tidal force(actually acceleration here), use this equation:
##a=GM\frac{d}{R^3}##
where a is the acceleration at the surface of the body being tidally distorted(say, the planet) along the line joining the two bodies. M is the mass of the source body(the satellite), d is the diametre of the body being distorted and R is the distance between the centres of the two. G is the familiar gravitational constant.

You can express it in units of e.g., Lunar tides. If the planet is the same size and mass as Earth, that lets you just say:
##a=\frac{M}{R^3}##
where R is the Andorlaine-satellite distance in units of Earth-Moon distance, M is the satellite mass in units of Lunar mass.
Multiply the result by 100 and you get it as a percentage of Lunar tides.
(to test it, you can plug in the data for the Sun, so as to find out how much stronger/weaker than Lunar are the Solar tides on Earth. Should come out as ~45%)

I'm not sure you quite got what the Hill sphere is all about. Or maybe I'm just missing some part of your reasoning.
Why did you calculate all those Hill spheres for the various aesthetic satellites? It's of no consequence. All you need, is the Hill sphere of the planet with relation to the star - it's the region around the planet where it is sensible to place satellites, as they can be expected to orbit the planet rather than just the star. Farther than that and the stellar gravity plucks the satellite from the planet's grasp.
Doing the same for those satellites, you find the radii at which an object would orbit those satellites. You don't need that.

Yeah, I get that... But too some extent, I don't have a true "feel" for the sort of number involved yet, so running the calcs for them was useful. (And also allowed me to catch a few more errors!)

And, as you say, it provides some useful boundary limits for me, so while I play with how big the moon actually is going to be (and it's apparent size), I've got some ideas to work with.

(It should also be noted that I sometimes do just play with numbers for the sake of it. I have approximately 1200 starships (plus those of my mates) for wargames across 40 starfleets in my set of starship rules... And a spread sheet whose sole purpose is to record the number of starships and I can list them by fleet points costs, fleet mass, tech level, cost per unit, cost per captial ship... All completely useless information in terms of game, aside from the pure meaningless joy of comparing numbers...!)

So that stuff and the inner/outer reach stuff, was, in part, familarising myself with what sort of numbers were involved and, as I say, getting a "feel" for it all.

Bandersnatch said:

How close it can be, is directly connected with the fact that the planet is tidally locked(or nearly so), and as such(at such great distance) will have very slow rotation.
How slow the rotation is, in turn, determines the evolution of the orbits of the planet's satellites. If the period of the (sat's)orbit is faster than the SLOW rotation of the planet, it will tidally interact with the planet in such a way so as to spiral down and eventually crash onto the surface*.

So, you need to calculate the geostationary orbit distance(satellite's orbital period=planet's rotation). This is the lower limit. The lowest orbit that won't see the satellite crash onto the planet some day.
But. It could take a LONG time, so take comfort in that. Think our Moon in reverse - it was once very much closer and it took it billions of years to move to where it is now, so you could just disregard it, unless you want to incorporate a doom prophecy into your writing or something.

(You should be able to do the calculations if you really need to. It's a secondary school-level exercise of comparing the gravitational force with the centripetal force; but do give us a shout if you end up stuck)

Gotcha. I can probably work that out, but if I'm struggling, I'll get back you. (Secondary school was twenty years ago - engineering degree was fourteen! - and I only have infrequent cause to do this sort of maths, so I'm a little rusty!)

Bandersnatch said:

What I meant to say when I first mentioned the Hill sphere, was that the geostationary orbit for a planet with a rotational period in the vicinity of a thousand+ years may well lie beyond the planet's gravitational influence(the Hill sphere).

Right. Still, nontheless, these sort of things are useful to know just to know (and when for comparison when I get the next bit done!). (I had not heard of the Hill sphere until you mentioned it, so I have been educated. I always like to say you learn something new every day!)

Bandersnatch said:

For tidal force(actually acceleration here), use this equation:
##a=GM\frac{d}{R^3}##
where a is the acceleration at the surface of the body being tidally distorted(say, the planet) along the line joining the two bodies. M is the mass of the source body(the satellite), d is the diametre of the body being distorted and R is the distance between the centres of the two. G is the familiar gravitational constant.

You can express it in units of e.g., Lunar tides. If the planet is the same size and mass as Earth, that lets you just say:
##a=\frac{M}{R^3}##
where R is the Andorlaine-satellite distance in units of Earth-Moon distance, M is the satellite mass in units of Lunar mass.
Multiply the result by 100 and you get it as a percentage of Lunar tides.
(to test it, you can plug in the data for the Sun, so as to find out how much stronger/weaker than Lunar are the Solar tides on Earth. Should come out as ~45%)

Cheers. I will have a good crack at that next time. (Which.. will probably next Monday, darn it, as things fall, since I usually use that as my quest-writing/world-building day, since I may have actual CADs work to do in the meantime...!)

How close it can be, is directly connected with the fact that the planet is tidally locked(or nearly so), and as such(at such great distance) will have very slow rotation.
How slow the rotation is, in turn, determines the evolution of the orbits of the planet's satellites. If the period of the (sat's)orbit is faster than the SLOW rotation of the planet, it will tidally interact with the planet in such a way so as to spiral down and eventually crash onto the surface*.

There are options:
If the satellite is small then the tides of the primary are weak and therefore the evolution of the satellite away from the geostationary orbit will be slow. It will then be still evolving after a long time. This is the case with both small satellites of Mars - Phobos and Deimos. They are still evolving, and have been evolving for a very long time.
If a satellite is evolving, then it crosses Roche limit well before actually reaching the surface of the planet. In which case it is likely to break into a ring.

On the other hand, if the satellite is massive, then the change in planet´s orbit also causes a significant change in the rotation of the planet. So if the satellite spirals down, it will speed up the rotation of planet so much that the planet becomes locked to the orbit of the satellite, and they come to a stable equilibrium before the satellite reaches the Roche limit. This has happened to Pluto and Charon.

So, you need to calculate the geostationary orbit distance(satellite's orbital period=planet's rotation). This is the lower limit. The lowest orbit that won't see the satellite crash onto the planet some day.

In case of small satellite in prograde orbit. Triton, for example, orbits slower than Neptune rotates - but in the other way, so is spiralling down.

What I meant to say when I first mentioned the Hill sphere, was that the geostationary orbit for a planet with a rotational period in the vicinity of a thousand+ years may well lie beyond the planet's gravitational influence(the Hill sphere).

Indeed. Imagine a satellite of Mercury or Venus. If it is inside the Hill limit, it will spiral down... but this might take a long time.

For tidal force(actually acceleration here), use this equation:
##a=GM\frac{d}{R^3}##
where a is the acceleration at the surface of the body being tidally distorted(say, the planet) along the line joining the two bodies. M is the mass of the source body(the satellite), d is the diametre of the body being distorted and R is the distance between the centres of the two. G is the familiar gravitational constant.

You can express it in units of e.g., Lunar tides. If the planet is the same size and mass as Earth, that lets you just say:
##a=\frac{M}{R^3}##
where R is the Andorlaine-satellite distance in units of Earth-Moon distance, M is the satellite mass in units of Lunar mass.
Multiply the result by 100 and you get it as a percentage of Lunar tides.
(to test it, you can plug in the data for the Sun, so as to find out how much stronger/weaker than Lunar are the Solar tides on Earth. Should come out as ~45%)

True.
But remember Kepler´s Third Law!
##{T^2}=\frac{R^3}{M}##
You can just substitute
##a=\frac{1}{T^2}##
If you are looking at the tidal acceleration of a satellite then it is completely independent of the distance to the primary or the mass of the primary. Moon experiences 81 times stronger acceleration due to Earth than Earth is experiencing due to Moon (same distance, different masses) but the tides which tiny Pluto causes in nearby Charon on a nearby 6 day orbit are close to the tides which huge Jupiter causes in distant Ganymede on its 7 day orbit.

But remember Kepler´s Third Law!
##{T^2}=\frac{R^3}{M}##
You can just substitute
##a=\frac{1}{T^2}##
If you are looking at the tidal acceleration of a satellite then it is completely independent of the distance to the primary or the mass of the primary. Moon experiences 81 times stronger acceleration due to Earth than Earth is experiencing due to Moon (same distance, different masses) but the tides which tiny Pluto causes in nearby Charon on a nearby 6 day orbit are close to the tides which huge Jupiter causes in distant Ganymede on its 7 day orbit.

This isn't correct here, is it? The M in one equation is not the same M in the other.
In Kepler's 3rd, M is the mass of the primary, which determines the orbital period of the secondary body(using the assumption of m<<M). When calculating tidal acceleration, M is the mass of the tide-inducing body, which in the case discussed is the secondary(we're talking about tides on the planet, not the satellite).

So, I'm having a play with the spread sheet and I'm struggling to get my head around it all.

To get a feel for it, I dropped the 85 AU box to 1 (to give me an Earth baseline), and changed rotation/revolution to 1/365 (0.0027), which gave 1 year for the Planet year, 0.0027 years for the sidereal rotation and -0.0027 years for the solar day. Which... doesn't sound quite right, as you said negatives should retrograde? (The Hill sphere is right at 0.01 AU.) The lowest stable moon orbit (full tidal lock) came out at 0.00028 AU (42000km), which is, obviously, less than the moon.

It appears from the spreadsheet that you can't get a value for the lowest stable moon orbit (full tidal lock) box that is inside the Hill Sphere for anything higher than 0.64189 (as far as I chased it) in the rotation/revolution. That value gave you 0.6419 years (about 234 days) per sidereal rotation and -1.8 years per solar day at 1 AU.

So at that limit, at 1 AU, you'd get three sidereal days to one solar day, in that case (and you'd have a solar day twice a long as a year).

Does that sound right?

It would also seem to suggest a tide-locked planet can never have a (stable) moon (let alone a tide-locked one). Yes? No?

(If yes, the question becomes "how stable is stable" in terms of "at what point does it consitute a time period of some hundreds of millions of years" between capture and release/impact.)

On the other part of the chart, the moon month length did not appear to have any effect on the tidal acceleration (only on mass, radius and distance), where as angular diameter had a huge effect. So for something to appear twice as big as the moon, it would have to as dense as a fairly light wood (450kg/m-3) to have the same sort of tidal forces (and month only set how big and far out.) That seemed a bit odd to me. A 50km radius moon made of chestnut or pine (closest wood density I could find) that appears twice the size of the moon orbiting at 33.6 hours per month has roughly the same tidal effect as Luna. Okay, yes. I can see that sounds about right, but it didn't really tell me much.

That all seemed a bit too circular (especially as I'm not sure what value of month to put in for a tide-locked moon on a tide-locked planet, assuming it's even possible) and as I was thinking about it and typing this up, I realised I've not asked quite the right questions.

I can use bits of your spread sheet to calculate tidal forces (I copied the tidal force in fractions of Luna formula and just plugged in the values I calculated above for a moon the size of Deimos that appears twice the size of Luna (moon radius= 6.277, orbital radius 1438.5km, mass 1.4762×1015) and that gave 0.38 Luna tides), so I can plug in whatever values I think I need (since month length doesn't seem to effect it, only mass and distance) but I realised than knowing the tidal forces (in Luna or otherwise) didn't tell me the answer I was looking for yet, which is at what point the tidal forces start turning the planet and how fast.

Extrapolating, tide-locking occurs when the tidal forces are equal, yes? So using the equations you provided earlier and a spread sheet1 I can set something up so that I can work out where aplanet =amoon, can't I? I.e.

##GMp\frac{dm}{R^3}=GMm\frac{dp}{R^3}##

Where Mp and Mm and dp and dm are mass and diameters of planet and moon (I couldn't figure out how to make the letters subscript without messing up the other tags.)

Yes?

If I set the diameter of the moon (and the distance because of the angular diameter), the major variable is the mass, which is going to be dependant on density.

Now, whether that will tell me something that is actually possible inside of the Hill sphere limits I don't know!

Maybe I'm looking at this the wrong way entirely. If I wanted a tide-locked "moon" maybe I could look at what would essentially be a second planet (or dwarf planet) tide-locked to the star, just slightly further out (i.e. so that the night side would get the "full moon") and orbiting very closely in a position and speed that would place it basically in the same place relative to Andorlaine. (I mean, thinking about it, that's sort of what you get with a tide-locked moon to a tide-locked planet isn't it?) Or, I suppose, even a more distant (but still awfully close by orbit standards) gas giant. In that situation, I would surmise that the two would have to be outside each other's Hill radius (or probably just on the very edge, on the assumption that for them to be in a "stable in the sense of a few billion years" orbit, they'd probably have to be some graviational attraction!)

(And I could have a smaller orbiting moon pretty much as I like if I make it small enough that the tidal forces are really small, if I can find a good balance between visible size, mass and distance.)

I'm getting there slowly. If nothing else, this is an education!

I did spend an hour or two on Saturday convincing myself of the luminosity figures and working out that if we go by R Coronae Borealis as the baseline, that Andorliane will get about 1425 W/m² and that (assuming a greenhouse effet => Earth) will be an average of 2º warmer.

I'm sort of tempted to say that the sun could do with being a bit cooler (extrapolating from the fact that red dwarfs have a higher infrared output and the red shift as a star cools: it'd deliver the same level of luminosity to the planet, it'd just be shifted a bit more towards the infrared end of the spectrum) - mostly for aethetic purposes (green plants is being a bit boring!) - but that may be taking a few too many liberties on RCB stars (especially since I don't think there's anything to compare it too.)

1WHY did I not think of plugging all this into a spread sheet instead of using my calculator, I do not know! It's not like I haven't used spreadsheets for this sort of thing before! *facepalm* Saves so much messing around!

It appears from the spreadsheet that you can't get a value for the lowest stable moon orbit (full tidal lock) box that is inside the Hill Sphere for anything higher than 0.64189 (as far as I chased it) in the rotation/revolution. That value gave you 0.6419 years (about 234 days) per sidereal rotation and -1.8 years per solar day at 1 AU.

So at that limit, at 1 AU, you'd get three sidereal days to one solar day, in that case (and you'd have a solar day twice a long as a year).

Does that sound right?

Sounds right - and demonstrates the absurdity of Hill "sphere". Look at the outer irregular satellites of Jupiter - both the prograde and retrograde ones. The stability conditions for outer satellites do NOT match the Hill "limit" and are not independent on the inclination or even direction of the orbit.

It would also seem to suggest a tide-locked planet can never have a (stable) moon (let alone a tide-locked one). Yes? No?

(If yes, the question becomes "how stable is stable" in terms of "at what point does it consitute a time period of some hundreds of millions of years" between capture and release/impact.)

Precisely. Just imagine Deimos orbiting Mercury. It would not be perturbed by Sun (too close to planet to matter). Sure, it would be slowly approaching Mercury - even more slowly than it is receding from Mars, because Mercury does not have atmosphere. This does not mean it could not spend more than the life of Solar System very slowly spiralling in. The tiny tidal torque that tides due to Deimos exert on Mercury would not come anywhere close to breaking the huge tidal torque locking Mercury to Sun. Yet on the other hand, the tides on Deimos would be overwhelmingly those due to Mercury - precisely because of "Hill" limit - and therefore Deimos would be tidelocked to Mercury.

On the other part of the chart, the moon month length did not appear to have any effect on the tidal acceleration (only on mass, radius and distance), where as angular diameter had a huge effect. So for something to appear twice as big as the moon, it would have to as dense as a fairly light wood (450kg/m-3) to have the same sort of tidal forces (and month only set how big and far out.) That seemed a bit odd to me. A 50km radius moon made of chestnut or pine (closest wood density I could find) that appears twice the size of the moon orbiting at 33.6 hours per month has roughly the same tidal effect as Luna. Okay, yes. I can see that sounds about right, but it didn't really tell me much.

but I realised than knowing the tidal forces (in Luna or otherwise) didn't tell me the answer I was looking for yet, which is at what point the tidal forces start turning the planet and how fast.

Extrapolating, tide-locking occurs when the tidal forces are equal, yes?

Comparing tidal forces shows which is the tidal lock to. It does not tell whether a tidal lock happens. There are 3 options - tidal lock between planet and satellite, tidal lock to Sun, or no tidal lock and free rotation instead.

Maybe I'm looking at this the wrong way entirely. If I wanted a tide-locked "moon" maybe I could look at what would essentially be a second planet (or dwarf planet) tide-locked to the star, just slightly further out (i.e. so that the night side would get the "full moon") and orbiting very closely in a position and speed that would place it basically in the same place relative to Andorlaine. (I mean, thinking about it, that's sort of what you get with a tide-locked moon to a tide-locked planet isn't it?) Or, I suppose, even a more distant (but still awfully close by orbit standards) gas giant. In that situation, I would surmise that the two would have to be outside each other's Hill radius (or probably just on the very edge, on the assumption that for them to be in a "stable in the sense of a few billion years" orbit, they'd probably have to be some graviational attraction!)

To get a feel for it, I dropped the 85 AU box to 1 (to give me an Earth baseline), and changed rotation/revolution to 1/365 (0.0027), which gave 1 year for the Planet year, 0.0027 years for the sidereal rotation and -0.0027 years for the solar day. Which... doesn't sound quite right, as you said negatives should retrograde?

Yeah, I got that backwards for the apparent path of the star on the sky. Negative values is E->W, positive is the other way around. Otherwise it looks good.

It appears from the spreadsheet that you can't get a value for the lowest stable moon orbit (full tidal lock) box that is inside the Hill Sphere for anything higher than 0.64189 (as far as I chased it) in the rotation/revolution. That value gave you 0.6419 years (about 234 days) per sidereal rotation and -1.8 years per solar day at 1 AU.

So at that limit, at 1 AU, you'd get three sidereal days to one solar day, in that case (and you'd have a solar day twice a long as a year).

Does that sound right?

It would also seem to suggest a tide-locked planet can never have a (stable) moon (let alone a tide-locked one). Yes? No?

Yeah, that sounds about right. And yeah, I don't think it's possible, outside having the moon be in L1 lagrangian point(which is unstable anyway, so there).

(If yes, the question becomes "how stable is stable" in terms of "at what point does it consitute a time period of some hundreds of millions of years" between capture and release/impact.)

I won't attempt to calculate it, as that's beyond my meagre abilities, but we can always go by what we see with our Moon. It is estimated that in the past the recession rate was less than 2cm/year(being lower than today's due to monocontinental landmass posing less of an obstruction to the moving oceanic bulge). If I understand the interactions correctly, it would be less for a lower mass satellite, as the bulge would be less pronounced(unless it cancels out with the lower inertia - don't know!). I haven't got a clue how much less would that be, but let's try and assume it is a linear relationship, so that a 10% mass satellite means 10% slower recession.

This should allow us to estimate the lifetime of the satellite(should be easy to plug into the spreadsheet). It will be most certainly inaccurate, but a ballpark figure is better than nothing.
Even bare timeframe calculations as if the mass doesn't matter will tell you someting.

I would expect at least hundreds of millions of years for anything not extremely close
.

On the other part of the chart, the moon month length did not appear to have any effect on the tidal acceleration (only on mass, radius and distance), where as angular diameter had a huge effect. So for something to appear twice as big as the moon, it would have to as dense as a fairly light wood (450kg/m-3) to have the same sort of tidal forces (and month only set how big and far out.) That seemed a bit odd to me. A 50km radius moon made of chestnut or pine (closest wood density I could find) that appears twice the size of the moon orbiting at 33.6 hours per month has roughly the same tidal effect as Luna. Okay, yes. I can see that sounds about right, but it didn't really tell me much.

It should be ~500km, not 50km, but otherwise it's O.K.
The orbital period doesn't change anything tidal acceleration-wise, as it is a variable that cancels out in the calculations - as long as the angular size is constant, increasing the period means placing the orbit farther, which required increasing radius to keep the angular size, which increases mass, which increases tidal acceleration by the same amount as it is lowered due to the increase in distance.

That all seemed a bit too circular (especially as I'm not sure what value of month to put in for a tide-locked moon on a tide-locked planet, assuming it's even possible)

The value in that case ought to be equal to that of planet rotation, only converted to days.

I can use bits of your spread sheet to calculate tidal forces (I copied the tidal force in fractions of Luna formula and just plugged in the values I calculated above for a moon the size of Deimos that appears twice the size of Luna (moon radius= 6.277, orbital radius 1438.5km, mass 1.4762×1015) and that gave 0.38 Luna tides), so I can plug in whatever values I think I need (since month length doesn't seem to effect it, only mass and distance) but I realised than knowing the tidal forces (in Luna or otherwise) didn't tell me the answer I was looking for yet, which is at what point the tidal forces start turning the planet and how fast.

Just to make sure you've noticed, the result is unphysical, as the orbital radius is below the surface of the planet(~6370km = Earth's).

Up to now, we've been disregarding tidal effect of the satellite on the planet. As long as the satellite is tiny, the effects are negligible, but they never dissapear.

Extrapolating, tide-locking occurs when the tidal forces are equal, yes?

No. It occurs when all the rotational angular momentum of one of the components has been dissipated by tidal forces. In the case of Earth and Moon, the latter has already tidally locked, but the Earth is still long way away from being locked, due to its high moment of inertia and difference in tidal force(Earth's on Moon>Moon's on Earth).

So using the equations you provided earlier and a spread sheet1 I can set something up so that I can work out where aplanet =amoon, can't I? I.e.

##GMp\frac{dm}{R^3}=GMm\frac{dp}{R^3}##

Where Mp and Mm and dp and dm are mass and diameters of planet and moon (I couldn't figure out how to make the letters subscript without messing up the other tags.)

Yes?

For LaTeX subsripts use "_". E.g., R^3_p nets you ##R^3_p##

Again, no. Yes, it tells you under what conditions the tidal accelerations are going to be equal, but it's of no consequence to your setup.

Maybe I'm looking at this the wrong way entirely. If I wanted a tide-locked "moon" maybe I could look at what would essentially be a second planet (or dwarf planet) tide-locked to the star, just slightly further out (i.e. so that the night side would get the "full moon") and orbiting very closely in a position and speed that would place it basically in the same place relative to Andorlaine. (I mean, thinking about it, that's sort of what you get with a tide-locked moon to a tide-locked planet isn't it?) Or, I suppose, even a more distant (but still awfully close by orbit standards) gas giant. In that situation, I would surmise that the two would have to be outside each other's Hill radius (or probably just on the very edge, on the assumption that for them to be in a "stable in the sense of a few billion years" orbit, they'd probably have to be some graviational attraction!)

Again, this doesn't really work. Being farther in orbit means slower orbital period(Kepler's laws), which won't hold the second planet in one spot in the sky.

(And I could have a smaller orbiting moon pretty much as I like if I make it small enough that the tidal forces are really small, if I can find a good balance between visible size, mass and distance.)

That's the general conclusion that I think is worth retaining.

I did spend an hour or two on Saturday convincing myself of the luminosity figures and working out that if we go by R Coronae Borealis as the baseline, that Andorliane will get about 1425 W/m² and that (assuming a greenhouse effet => Earth) will be an average of 2º warmer.

I'm sort of tempted to say that the sun could do with being a bit cooler (extrapolating from the fact that red dwarfs have a higher infrared output and the red shift as a star cools: it'd deliver the same level of luminosity to the planet, it'd just be shifted a bit more towards the infrared end of the spectrum) - mostly for aethetic purposes (green plants is being a bit boring!) - but that may be taking a few too many liberties on RCB stars (especially since I don't think there's anything to compare it too.)

From that paper I linked to in post #2, it appears that you can have as high as 2200 W/m^2 on a tide-locked planet without significanly altering the temperature. Admittedly, it only considered red dwarf stars, but it does give you a bit of a leeway in the artistic liberties department, I think.

1WHY did I not think of plugging all this into a spread sheet instead of using my calculator, I do not know! It's not like I haven't used spreadsheets for this sort of thing before! *facepalm* Saves so much messing around!

I know, right? Had the same revelation a while ago :)
By the way, if you can get your head around the equations, I advise to make your own spreadsheet from scratch. It's hard enough to remember your own thought processes behind the formulas you put in there at 1 AM after a couple of beers. Doing the same with other people's is a daunting task indeed.

Yeah, I got that backwards for the apparent path of the star on the sky. Negative values is E->W, positive is the other way around. Otherwise it looks good.

Right! I thought it might be something like that, but I wasn't sure.

Bandersnatch said:

I won't attempt to calculate it, as that's beyond my meagre abilities, but we can always go by what we see with our Moon. It is estimated that in the past the recession rate was less than 2cm/year(being lower than today's due to monocontinental landmass posing less of an obstruction to the moving oceanic bulge). If I understand the interactions correctly, it would be less for a lower mass satellite, as the bulge would be less pronounced(unless it cancels out with the lower inertia - don't know!). I haven't got a clue how much less would that be, but let's try and assume it is a linear relationship, so that a 10% mass satellite means 10% slower recession.

This should allow us to estimate the lifetime of the satellite(should be easy to plug into the spreadsheet). It will be most certainly inaccurate, but a ballpark figure is better than nothing.
Even bare timeframe calculations as if the mass doesn't matter will tell you someting.

I would expect at least hundreds of millions of years for anything not extremely close

Precisely. Just imagine Deimos orbiting Mercury. It would not be perturbed by Sun (too close to planet to matter). Sure, it would be slowly approaching Mercury - even more slowly than it is receding from Mars, because Mercury does not have atmosphere. This does not mean it could not spend more than the life of Solar System very slowly spiralling in. The tiny tidal torque that tides due to Deimos exert on Mercury would not come anywhere close to breaking the huge tidal torque locking Mercury to Sun. Yet on the other hand, the tides on Deimos would be overwhelmingly those due to Mercury - precisely because of "Hill" limit - and therefore Deimos would be tidelocked to Mercury.

Okay. Should be able to take a stab at that, then, once I've ascertained the right moon size/distance (etc).

Bandersnatch said:

Up to now, we've been disregarding tidal effect of the satellite on the planet. As long as the satellite is tiny, the effects are negligible, but they never dissapear.

I think we can live with the planet not being completely tide-locked: if the tidal force is small enough that the planetary rotation is in thousands of years, it may as well be tide-locked as far as the civilisations go. (And for the whole orbital system, "stable" in terms of "long enough for life to utilise the moon/ long enough to be measured in terms of the rise and fall of civilisations" is plenty stable enough, even if it's not stable by stellar terms.)

Bandersnatch said:

For LaTeX subsripts use "_". E.g., R^3_p nets you ##R^3_p##

Aha. Cheers.

Bandersnatch said:

No. It occurs when all the rotational angular momentum of one of the components has been dissipated by tidal forces. In the case of Earth and Moon, the latter has already tidally locked, but the Earth is still long way away from being locked, due to its high moment of inertia and difference in tidal force(Earth's on Moon>Moon's on Earth).

Again, no. Yes, it tells you under what conditions the tidal accelerations are going to be equal, but it's of no consequence to your setup.

Right. Okay, then.

Bandersnatch said:

Again, this doesn't really work. Being farther in orbit means slower orbital period(Kepler's laws), which won't hold the second planet in one spot in the sky.

Right. Well, that's pretty clear - I'm gonna have to ditch the "moon on the night side (for the purposes of Lighting the Night Side A Bit)" idea completely, since it is apparently basically impossible.

Bandersnatch said:

That's the general conclusion that I think is worth retaining.

Small aesthetic moon it is then. So I need to make it small enough to minimise the tidal effects, but big enough so that the desired apparent magnitude actually puts it above the surface of the planet...!

(And the moon's orbital period then, will form the basic unit of time measurement (day/month/year, depending on exactly what the period ends up as!)

Still... If the moon is going to be as apparently visible as Luna on Earth or more so (and the moon has a high albedo as I'm intending), that would itself mean that it would light the night side anyway as part of it's orbit. Which... interestingly... would give the night side a sort of "day/night cycle" in terms of illumination.

I don't know if you'd get much heat, though. I can probably take an estimate of it, though, using the same set of calucalations used to derive the amount of energy delivered to the planet, couldn't I? It's just be a case of working out how much energy you'd get on the moon, using the inverse of the albedo, wouldn't it? (Because you're looking at what bounces off.) And then applying that to the planet?

Reason I'm thinking all that is, if you could get a reasonable reflection of energy off the moon, you might be able to have proper night-side plants that photosynthesise off moonlight. Which would be really kind of cool.

Bandersnatch said:

From that paper I linked to in post #2, it appears that you can have as high as 2200 W/m^2 on a tide-locked planet without significanly altering the temperature. Admittedly, it only considered red dwarf stars, but it does give you a bit of a leeway in the artistic liberties department, I think.

I was thinking more of the sun being cooler less for keeping the planet's temperature lower (the planet's temperature wil just inform me of the ecology!) than for the fact that the visible light would be more red-shifted (and thus a bit more... evening-y as well as making the plants not be green). But given that most of the civilisations are going to likely be towards the terminator, the sun's going to be permenantly sunset, so it's probably not that important.

Presumably, if a plant is in permenant evening light, it'd want to be a different colour (i.e, closer to the black that you'd theorhetically get on your dimly-lit tide-lcoked red dwarf world) to optimise the lesser amount of light, yes?

Bandersnatch said:

I know, right? Had the same revelation a while ago :)
By the way, if you can get your head around the equations, I advise to make your own spreadsheet from scratch. It's hard enough to remember your own thought processes behind the formulas you put in there at 1 AM after a couple of beers. Doing the same with other people's is a daunting task indeed.

Oh yeah. My first real attempt at a spreadsheet was for doing the points costs for my starship rules. Now, at this point, we've gor between me and my mates, about 47 starship fleets (some fleets use the same models out of our... 1600 starships (1200 of which are mine. You could say I have something of a starship obession...!)) so trying to do it by hand is just impractical (and when I did it, we must have had no less than half of that). And occasionall, as I narrowed down the bugs, I'd spend almost a long back-working what it was I'd done than working out the solution!

Okay, I have had a serious stab at things, first off, making my spread sheet! With a bit of looking (and notes to a calculator for getting magnitudes from luminosity and distances (because I couldn't get my head round the formulas, it was just easier!) I've got a good start.

What WAS interesting was me finally going back to read that paper on dwarf stars you linked early on, Bandersnatch. I wouldn't have understood it, I don't think, without messing around with the global temperatures last week. But what THAT said was that the planetary albedo on a tide-locked planet would actually be very high because of the clouds. So, far from Andorlaine being 2º warmer, it would be actually be 15 degrees colder . (More, actually since Earth greenhouse effects add +33º and if I undertstand that document right, at that it only only adds +5.6º)

So, I have experimented pushing Andorlaine to about 106AU, which gives 2250-ish W/m², and again, if I'm understanding the equations and data right, thatn gives the planet a mean temperature of about 280ºK (i.e. about 10ºC) (i.e. 261º+21º green house effect). (And that's unfortunately, at the limit of what the document goes with greenhouse effects and I am kind of completely lost as to how to derivive that myself... I've tried to get my head around the equations, but it just sort of blurs out. I can, I guess, try interpolating something another time. Mind you, these guys even used a computer simulator to get this data so...!)

(Also, this means that if I were to make the star cooler (to shift the light more to IR and red), Andorlaine would have to get a lot closer (as it's still arguably not close enough!))

From somewhere else, it suggests that the maximum temperature at the hot pole would be mean times √2, which in the case of the 106 AU planet would give you about 127º C (400ºK). The actual temperature would be lower than that (though I have no idea how you'd actually calculate it!)

Andorlaine from space will be dominated by a sunward cyclone. The upper atmosphere super rotates (like Venus). The lower atmosphere blows warm air at ground level from the day-side across the equatorial terminators, while cold air flows back to the day-side over the poles; thus the surface is subjected to continuous winds at about 20mph.

The global temperature is about +4ºC higher than Earth's, though temperature variation is likely to be more modally extreme over the surface.

The night-side has a large ocean, whose surface is frozen, but which contains liquid warer beneath. The sun-side is a more mountainous region, with heavy and continuous rain, and while it has some extremophile flora and fauna, there are few larger plants or animals able to survive there. The terminator between is most populated on the sunward side, where the plants get the most light. Flora closest to the subsolar point is more green, but darkens to black as it approaches the terminator, which is lit in a state of continuous evening. (The plants there are akin to what is found in Red Dwarf stars where luminosity is lower.)

Andorlaine's moon is a relatively recent capture. Composed almost entirely of water-ice, is thus has a high albedo. It appears significantly brighter than Luna, and the dim moonlight allows some limited, highly specialised and very slow-growing extremophile flora on the night side. The night-side thus paradoxically enjoys something of a day-night cycle, due to the six day orbital period of the moon.

Andorlaine is not quite completely tide-locked. Due to a combination of volcanicity (partly caused by orbital resonance of Andorliane's nearest naighboruing gas giant), dwindling angular momentum and the effect of the moon, the crust rotates almost imperceptibly, resulting in a day length of a few thousand years.

At irregular intervals with a period of 400-1100 years, Andorliane's primary dims significantly, dropping eight magnitudes in the space of a few months, due to excreted carbon dust. This period can last for up to a few hundred years (the time between dimming correlates to length of the dimming period). The precise reason Andorlaine's sun has dim periods considerably longer than other RCB variable stars, and why is has retained this stage much longer, is as yet unknown. Theories suggest exotic materials in the star are causing it to burn "slower" than a typical star. Other theories postulate that the ejected dust clouds are partical recycled by the distant orbit of the binary star companions of the primary.

During these periods, the sun is still visible, but is muted and much more red-shifted. Further, the amount of visible light reaching Andorlaine is slightly less than expected, making it appear even darker. The infrared radiation is lower, but not significantly so, so while the global climate does cool, it is a relatively mild and slow drop. This does causes a spread of the night-side flora across the terminator (and the ecology that follows) and a corresponding shift of the terminator flora towards the subsolar point.

The irregular frequency of these events is such that the native civilisations (of which there have been several) are often caught unawares, previous events being myth or legend. While the ecology of Andorlaine goes into minor upheaval during these events, primitive civilisations have all thus far been unable to cope with the additional pressures and collapse. Some of the more advanced civilisations (which arose in periods where the intervals between dimming were greatest) became aware of the phenomina, but had not reached a sufficient level to be able to adapt fast enough.

(Imagine the results of what would happen on Earth even with 21st century technology - which some of the civilisations approached - if the sun suddenly dimmed drastically.)

The moon was calcuated (eventually), based on Tethys in beaing mostly water-ice with a high albedo. Making a very small, irregular moon easily visible by the naked eye proved to be impractial, as you couldn't get it close enough without hitting the atmosphere - likewise, making one double the apparent size of Luna put considerable tidal acceleration on the planet. (Andorlaine's moon imparts 0.26 of Luna's tidal forces. I am assuming therefore a) the length of time it is captured will be in the same "billions of years" order of magnitude and b) there will be some tides in the sea, which will be much less noticable and possible more frequent.)

The sun being the same angular diameter was honestly more accident than design - that's just what it turned out at at the right distance for the solar flux (picked on mean global temperature) I wanted. (An averagely warmer-than-Earth planet seemed like it would stand the temperature shifts a bit better.)

The climate stuff is a bit more tenuous. The sources I looked at were, in some cases, reported results of a model, so is a bit more guesswork and extrapolation and I'm well aware that they could be wrong. (But then again, all of this is very theorhetical anyway - I have cme to realise I'm looking really at the bleeding edge of astrophysical theory! Still, let it not be said that I didn't make a spirited effort for plausibility!)

The numbers are not quite final yet - I will probably play with them a touch more to make them a bit more varied off the baselines (to make them a bit more "real" - beauty of a spreadsheet!), but I think this is my overall starter for ten, at least enough that I can spend my thinking time on holiday starting to consider what actually lives there.

Thoughts, feedback - especially on the more tenuous areas like climate, weather patterns and temperature I've been hazarding, or the sort of extrasolar flora/fauna that might arise are welcomed!

*Both these values are very, very approximate. They are pretty much eyeballed from a source that showed the minimum/maximum temperatures at up to 1400W/m², which were consistent with the sort of values as derived from the paper Bandersnath linked in post #2 (from which the greenhouse and albeo numbers are estimated), taking into account other estimation people had made which fell into a similiar range values, though I do not believe those calculations involved greenhouse effects. I thus haven't quite pulled them out of the ether, but I think they should be in the right sort of ballpark. Maybe. I have come to the conclusion though, that it may well be impossible to get the true answer without actually having some sort of appropriate computer model (since that's where the people who got their values got them!) I thus treat them as very dubious! (According to some other data, the absolute maximum for the hot pole would be √2 times the global mean temperature (which would be ≈371K), but that did not account for cloud formation.